Idempotent completion of certain $n$-exangulated categories

TL;DR

This paper proves that the idempotent completion of certain $n$-exangulated categories retains an $n$-exangulated structure, broadening the class of such categories beyond existing frameworks.

Contribution

It generalizes previous results by showing idempotent completions of $n$-exangulated categories are themselves $n$-exangulated, even when not $n$-exact or $(n+2)$-angulated.

Findings

Idempotent completion preserves $n$-exangulated structure.

Provides examples outside $n$-exact and $(n+2)$-angulated categories.

Extends the theory of $n$-exangulated categories.

Abstract

It was shown recently that an $n$-extension closed subcategory $\mathscr A$ of a Krull-Schmidt $(n+2)$-angulated category has a natural structure of an $n$-exangulated category. In this article, we prove that its idempotent completion $\widetilde{\mathscr A}$ admits an $n$-exangulated structure. It is not only a generalization of the main result of Lin, but also gives an $n$-exangulated category which is neither $n$-exact nor $(n+2)$-angulated in general.